The K-user multiple-input single-output broadcast channel is considered under per-antenna power constraints, i.e., each transmit antenna must satisfy its own power constraint. A low complexity zeroforcing (ZF) precoder is proposed when the number of transmit antennas M is greater than K. The proposed precoder design significantly reduces computational complexity for the precoder construction while attaining the sum spectral efficiency close to that achievable by the optimal ZF precoder.